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This paper provides a conceptual analysis of the role of the mathematical continuum versus the discrete in the understanding of randomness as a notion with a physical meaning or origin. The presentation is ‘informal’ as we will not write formulas; however, we will refer to non-obvious technical results from various scientific domains, and we will also propose a conceptual framework for understanding randomness (and predictability), which we believe is, essentially, original. As a matter of fact, unpredictability and randomness may be conveniently identified in various physico-mathematical contexts. This will allow us to explore these concepts in continuous versus discrete frameworks, with particular emphasis on the relationships and differences between classical approaches and quantum theories in Physics.